In the paper reviewed above the authors characterized the composition of high scoring segments among partial sums of i.i.d. random variables. In this paper the authors consider a letter sequence assuming values from a finite alphabet governed by an -state irreducible Markov chain. Suppose are independently distributed given the sequence , where the joint distribution of depends only on and and is of bounded support. When is started with its stationary distribution, and the existence of a finite cycle such that is assumed. Define the stopping times
and for ,
Theorems 1 and 2: Let while . Then a.s. as , and a.s. as , for any value of , certain positive constants , .